Model selection via marginal likelihood estimation by combining thermodynamic integration and gradient matching

Conducting statistical inference on systems described by ordinary differential equations (ODEs) is a challenging problem. Repeatedly numerically solving the system of equations incurs a high computational cost, making many methods based on explicitly solving the ODEs unsuitable in practice. Gradient matching methods were introduced in order to deal with the computational burden. These methods involve minimising the discrepancy between predicted gradients from the ODEs and those from a smooth interpolant. Work until now on gradient matching methods has focused on parameter inference. This paper considers the problem of model selection. We combine the method of thermodynamic integration to compute the log marginal likelihood with adaptive gradient matching using Gaussian processes, demonstrating that the method is robust and able to outperform BIC and WAIC.